13.1 Using General Relativity to Change the Remote Coordinate Speed of Light c'

In the physics of special relativity theory, travel faster than the vacuum speed of light must be forbidden in order to preserve causality(effect never precedes cause). And in the physics of special relativity theory the vacuum speed of light is globally c (Eqn. 1.1.7 c = 299792458m/s exact by definition). Therefor in the physics of special relativity theory there is no faster than c travel. However, in the physics of general relativity theory the vacuum speed of light is only locally invariant and so the remote coordinate speed of light is not always c, and in some cases it is greater than c. There are metrics as shown below for which the coordinate speed of light c' can be increased above c. Because of this there are situations in the physics of general relativity theory where objects travel faster than c.

Lets consider for a moment the invariant interval for the theory of general relativity for the world line of a particle moving in the x1 direction, and dx1 is a coordinate distance displacement.

ds2 = g00dct2 + 2g01dctdx1 + g11dx1dx1

(13.1.1)

If this is a photon, then ds2 = 0.

0 = g00dct2 + 2g01dctdx1 + g11dx1dx1

0 = g00 + 2g01(dx1/dct) + g11(dx1/dct)2

0 = (1/2)g00 + g01(c'/c) + (1/2)g11(c'/c)2

So the remote coordinate speed of light for the theory of general relativity is given by,

So for general relativistic physics we see that objects can travel faster than c in the + x1 direction if

{-g01 + [(g01)2 - g00g11]1/2}/g11 > 1

(13.1.3)

Exercises

Problem 13.1.1

Consider light instantaneously undergoing a coordinate angular displacement in a spacetime described by

Show that angular moving light moves at

Problem 13.1.2

Out in the depths of a special relativistic spacetime a wheel space station of radius r0 rotates to produce artificial gravity so that the rim rotates at a speed of

An observer onboard in the rim centers a cylindrical coordinate system upon himself so that the center of the wheel is always at

The exact solution for the invariant interval according to an appropriate choice for this observer's coordinates is:

a. Consider a clock at the center of the wheel yielding time t. Show that this spacetime yields

b. Consider a light-like signal pulse that travels from the observer (strait according to the inertial frame of the center clock) to the center and is reflected so that it returns to the observer. Show that:

c. The time for the round trip according to the inertial frame according to which the central clock is at rest is

The time for the round trip according to the observer's frame who is in the rim of the wheel must be

Why must this also yield ?

d. Show that local to the observer the spacetime approaches the form for an inertial cylindrical frame and that the coordinate speed of light approaches c.

e. Show that in a sufficiently Newtonian limit, that the geodesic equation yields motion consistent with all of the fictitious forces predicted for this scenario by Newtonian physics.

f. Show that the limit as r0 goes to zero yields

This would be the spacetime according to an observer merely spinning on his own axis.

To the extent that we restrict our physics to special relativity theory and demand that the traditional order of causation is always correct, no information can travel faster than the speed c. However, the theory of special relativity is only a special case of physics within a more general theory known as the theory of general relativity.

In general relativistic physics the vacuum speed of light is everywhere locally invariant, but not globally invariant. The vacuum speed of light remote from a given observer need not be c according to that observer's remote coordinates. It can be greater than c and can vary with direction depending on the gravitational field involved. Because this is allowed, starships that travel between the stars faster than c are not ruled out a-priori within general relativistic physics.

In 1994 physicist Miguel Alcubierre of the University of Wales published [Alcubierre, M., 1994] how to use general relativistic theory to consider a space-time geometry, actual warp drives, that would undo the special relativistic time dilation effects and allow for faster than c travel in the May issue of Classical and Quantum Gravity. His space-time geometry would also allow for acceleration of the ship without any internal forces crushing the crew. They would feel weightless as the ship accelerated. Unfortunately in the arbitrary manner that the metric was developed it was shown by Pfenning & Ford [Ford, L. H., M. J. Pfenning, 1997] to require a lot of negative energy that would violate a few quantum mechanics energy conditions and a greater magnitude of it than all the mass of the known universe. This section will demonstrate a way to arbitrarily lower the magnitude of the negative energy requirement.

A case of the space-time geometry that he had derived can be represented in the following equation

(13.2.1)

f can be any function of the coordinates that is one at the location of the starship and zero far from it. Transforming coordinates to a particular choice of the starship coordinate frame and allowing it to travel faster than c , coordinate singularities crop up in the equation corresponding to event horizons in front of and behind the ship enclosing it in a mathematical bubble. This is called the warp bubble. Even though the event horizons remain for any choice of ship frame, there is a choice for star ship coordinates for which the singular nature is transformed away.

June 1999 Chris Van Den Broeck of the Institute for Theoretical Physics at the Catholic University of Leuven, Belgium, came up with an alteration [Broeck, C. V. D., 1999] for this space-time geometry that would retain all of the desired warp drive qualities but reduced the negative energy requirements down to the order of a transversable wormhole. His space-time geometry can be represented by this equation

(13.2.2)

B can be any function that is large near the starship and 1 far from it though he used a specific top hat function for his example calculations. That not only brought the negative energy requirements down to a hopefully one day reachable goal but it also solved one of the quantum mechanics energy condition violations.

The above metrics for general relativistic physics were designed to study linear motion and Alcubierre himself wrongly believed that the effect of the acceleration beyond c speeds of the Starship was a result of the space behind the ship happening to be in a state of expansion while the space in from of the ship is in a state of contraction. This isn't really the cause. There are other Warp drive spacetimes besides the one above. Consider the following warp drive loop.

(13.2.3)

Let be any function of time. Let f be a function that is zero far from the warp loop and is at a maxima of one at (z = 0, r = R). This spacetime has a geodesic at the loop such that an object initially following the loop at its initial warp speed will continue on accelerating according to however is changed.

Notice that in this spacetime for general relativistic physics there is no diametric nature to the space immediately behind and in front of the ship, thus spacetime expansion and contraction is not the causal factor for these kinds of warp drives.

The Van Den Broeck Warp Drive of general relativity was actually a case of a more general type of Alcubierre-Broeck Warp Drive SpaceTime. A more general case of an Alcubierre-Broeck Type Warp Drive SpaceTime for general relativistic physics can be written

(13.2.4)

Where the Metric tensor reduces to that of special relativity far from the ship, and dx1 represents a coordinate distance displacement.

The transformation of coordinates to the ship frame and the interval in the ship frame were given by Hiscock for the case of constant velocity in the consideration of only two dimensions [Hiscock, W. A., 1997]. We here show how his transformation extends to four dimensional spacetime with arbitrarily time dependent acceleration. We also present the ship frame energy density T00 from a four dimensional calculation and note that the 4d classical calculation is everywhere finite.

Consider an Alcubierre interval for general relativity given according to a remote frame's cylindrical coordinates by

(13.2.5)

Where f is a function that is 1 at the location of the ship and zero far from it.

We start out with the first transformation Eqn 13.2.7

Where is first expressed here as a function of time ct.

With some algebra for simplification this results in

Let g = 1 - f and this becomes

(13.2.6)

Notice that this returned the original intervals form with a reversal on the sign of and a reversal of the boundary conditions for g.

Now we notice that at (z,r) = (0,0), this interval becomes the interval for special relativity transformed to cylindrical coordinates. Thus, we have found a transformation to a frame based local to the ship. One can also verify that in these coordinates the relevant affine connections vanish at (z,r) = (0,0).

Therefor this interval works not only for a frame local to the ship, but also for ship frame itself. So the global coordinate transformation between the remote frame and the ship frame is

(13.2.7)

Using and , based on this interval, the ship frame energy density works out to be

(13.2.8)

The total ship frame energy for the general relativistic physics is

(13.2.9)

(13.2.10)

Other terms in the stress-energy tensor involve and so that this doesn't diverge at the boundaries leaving a delta function matter distribution there we can also require to be continuous at the boundaries. If the warped region is modeled as a spherical shell of inner radius R and outer radius then we might choose for example calculation:

Compare to example calculation of Pfenning and Ford Class. Quantum Grav. 14 (1997) pg 1750 equation 28, the difference being in the choice of behavior for g. However, dimensions between the extremes do NOT result in more energy magnitude requirements than contained in the universe. The calculation that led to that result was an extreme limit where . This limit was required by the assertion that a quantum inequality restriction due to the presence of negative energy matter limited the thickness of the warp shell (the matter responsible for the warp).

For an example calculation take, traveling 10 times the speed of light. Take R = 10m, . At this point, these dimensions are not really allowed by a quantum inequality due to the presence of the negative energy, but we later show how to even further reduce the negative energy result which then puts these dimensions within the limit allowed by the quantum inequality. These inputs result in a total energy requirement of

Absolutely not the ridiculous large claims that it must be on the order of magnitude of the mass of the universe.

The reasons for the difference are

1.) They have restricted their warped region's thickness by a quantum inequality where we have not. We do not as we later show how this negative energy can be arbitrarily reduced until any particular dimensions are allowed by the inequality.

2.) We have chosen a much different behavior for the function g between the boundaries than Alcubierre had originally chosen, and we looked at an intermediate case for the extremes of R and . We could consider speeds greater than c in which we must still address other issues such as the locally tachyonic motion of the matter at the outer edge of the warp, but at speeds just under c the only real showstopper is the weak energy condition violation. Fortunately there are ways in which the weak energy condition can be violated in nature such as production of Casimir effect energy, or otherwise lowering the zero point energy of the vacuum.

Instead of using, t' = t, IFis a constant, say , and IFwe consider only two dimensions (ct,z), then the spacetime for general relativity may be diagnalized by doing the next transformation

(13.2.13)

Then we arrive at Hiscock's interval

ds2 = H(z')dct"2 - dz'2/H(z')

(13.2.14)

Now we see that singularities associated with the event horizons occur in the invariant interval where H(z') = 0, but these singularities only exist with such a particular choice of time for ship frame coordinates. With our previous choice of time coordinates these singularities were transformed away and using that frame we don't have problems with divergence of the energy density.

Next we will go back to the ship frame Alcubierre metric of general relativity Eqn.13.3.7

Alcubierre's original more general metric had a time dilation term in the remote observer's frame

. I call it a time dilation term because of how it is related to U0 in Eqn.13.2.19 below. Alcubierre prefers to call it by the lapse function. Instead, we will reintroduce such a term into the ship frame's interval, and we will use different boundary conditions for it than Alcibierre did. We will keep A = 1 both at the location of the ship, and far from it, but allow it to become large in the warped region. The interval in the ship frame becomes

(13.2.15)

The solution for T00 according to the ship frame is

(13.2.16)

It is the ship frame calculation of the energy density that is important here because it is the ship that provides the energy, and now we see that T00 can be arbitrarily lowered by making A arbitrarily large in the warped region.

Another concern over the Alcubierre warp drive of general relativity is that it once the ship goes superluminal, there may be a problem controlling the warp to turn it off and slow the ship. At superluminal speeds, there are event horizons for this spacetime where

g00 = 0, even in the case that we have transformed away the singularities there. These two horizons can be constructed as two half spheres enclosing the ship in a warp bubble in general relativity. Information can not be sent from behind the ship outside the bubble to the inside. Also, information can not be sent from inside the ship to the region to outside, in front. For the warp drive, part of the matter region producing the warp, or the warp shell extends across the horizons. A signal sent from inside the ship can not reach the matter extending in front of the horizon that is in front of the ship. This piece of the warp shell can not then be turned off. This lead to the concern over the ability to control the warp at superluminal speeds. We can work around the problem as follows. The matter controlling the behavior of A(ct,xi ) can be arranged prior to the matter controlling the behavior of .

It is the behavior of and not A(ct,xi ) that controls the speed of the ship. The horizon occurs where

The function g is then manipulated so that it goes to 1 at a smaller distance from the ship then where

.

In other words, the function A(ct,xi ) is larger than for an interval extending beyond the place where g becomes 1. Now the portion of the matter shell that was formed to control the behavior of A(ct,xi ) that is in front of the horizon will not be accessible by a signal from the ship once superluminal speeds have been reached, but as far as controlling the speed of the ship goes, this does not matter. What matters is that the portion of the matter shell that controls the behavior of is totally contained inside the horizons ensuring that the ship speed is controllable even after the ship has gone superluminal.

Finally in investigation of the new quantum inequalities, we return to the Alcubierre spacetime for general relativistic theory according to the ship frame modified by inserting into the ship frame's metric

Pfenning applied a quantum inequality [Ford, L. H., M. J. Pfenning, 1998] for a free, massless scalar field to the alcubierre warp even though he did not include the possibility of an other than A = 1, and even though the Alcubierre spacetime for general relativity is not really the result of a free, massless scalar field. Even so, we can go back and redo the calculation along the same lines with a variable included. The quantum inequality he applied which we recalculate is

(13.2.17)

Alcubierre claimed his energy density calculation was based on a Euclerian observer, however, it was really based on a ship frame observer, who he called Euclerian, one who was in the ship, moving with it, and with our modification of where A is inserted and its boundary conditions, in order to apply it we must find and for a true Euclerian observer, one who samples the energy density where it passes him. The correct Euclerian observer for this general relativistic physics problem is an observer who "samples" the time it takes for some flux of negative energy to pass him, and has such a velocity that he comes to rest with the ship as the ship overtakes him.

This Euclerian observers for this modified spacetime interval will be those with a velocity along the z' axis given by

(13.2.18)

so the interval we have

Inserting U we wind up with

Again, this result is why I term the lapse function a time dilation term.

then makes an order of magnitude aproximation of the geodesic motion of the Euclerian observer. Equation (5.11). Since we are using the ship frame with inserted, we must include its effect here as well.

We make a definition corresponding to Pfenning's equation 5.15

(13.2.20)

And choose corresponding to his choice for 5.4

Consider A kept constant with respect to z, and therefor , through the negative energy region. According to Pfening's 5.16 the integral is approximately

This result is the same as Pfenning eq 5.17 with the exception that there is an A(r) is not necessarily 1. Pfenning then asserts that this result must hold for sample times that are small compared to the square root inverse of the largest magnitude of the Riemann tensor components as calculated for the local observers frame.

If for simplicity A is kept approximately constant through the negative energy region, then the largest Riemann tensor component for this spacetime is

(No sum on which is only the index for the ship frame radial distance, not a variable index.)

So in this case, the sampling time must be restricted to

where .

Inserting this into the inequality and looking at the case of large A approximately constant A0 through the negative energy region leads to

Choosing , and making the approximation , this becomes

(13.2.23)

Now we see that letting A become arbitrarily large also arbitrarily thickens the minimum warp shell thickness for general relativity. Therefor the calculation had a reachable shell thickness. All that remains then is to divide this result by an A04 which would allow the thickness chosen, and which will lower the energy magnitude even farther by several orders of magnitude.

Exercises

Problem 13.2.1

Let A be approximately constant throughout the region that g varies. What does A0 have to be to allow the chosen shell thickness in the example energy calculation of ? What does this reduce the energy calculation to?

Problem 13.2.2

Use a computor program such as grtensor II, easily found on Google, to find the remote observer coordinate frame solution for the stress energy tensor for the spacetime of equation 13.2.3 by calculating the Einstein tensor to show that the stress energy tensor is given by

We have seen that the remote coordinate speed of light for light moving in the direction for general relativistic physics is given by Eqn.13.1.2

when dx1 is a coordinate distance displacement. We can use this to make faster than c travel possible in two ways. The first way we will demonstrate results in a general Alcubierre-Broeck type of warp drive space-time. The second results in a new type of warp drive.

First, consider a starship moving along a geodesic in the x1 direction with a speed v given by

. Since this speed is never outside of the boundaries of the speed of light given above, we can allow any speed, even speeds greater than c. So consider a spacetime where

at the location of the ship and far from it. Let f be any function that is 1 at the location of the ship and 0 far from it. We can then do this by requiring

.

Then g00 can be chosen to be which ensures that the remote coordinate speed of light is globally real.

This leads to a general Alcubierre-Broeck type of warp drive space-time given by Eqn.13.3.4

where dx1 is a coordinate distance displacement. Here, f is any function that is 1 at the location of the ship and 0 far from it. A is a function of the coordinates that is 1 far from the ship. The other metric components must also reduce to those of special relativity far from the ship.

For the interval between events along the world line of the ship , where A0 is A evaluated at the location of the ship.

The second way to go about achieving faster than c travel along these lines goes as follows. Consider the case of a diagonal metric.

The coordinate speed of light is then given by

This speed of light is greater than c when . This leads us to a new type of warp drive space-time.

Consider the following interval expressed in a remote observer's coordinates

(13.3.1)

Far from the ship

T = Z = Y = X = 1

The coordinate speed of light moving along the z direction is

(13.3.2)

This is the new maximum speed for objects in motion in the z direction. Recall how in section 12.4 we patched together different vacuum field solutions through a region of a spherical shell matter. Here we will do the same thing. We will use two concentric matter shells to patch together three vacuum field solutions. Out side of the outer shell the interval in the external observer's coordinates will be a vacuum field solution that reduces to far from the ship.

For the vacuum region between the outer and inner shells the interval will be given by

(13.3.3)

Also to avoid local tachyonic motion of the matter in creating the disturbance in the metric, we will let the the metric tensor at the outer edge of the outer shell equal the metric evaluated at the outer edge of the inner shell. Let single primes represent the ship frame coordinates. The transformation between the external observer's coordinates and the ship frame coordinates is a composition of the two transformations that results in

From these we find the velocity transformation equation for motion in the z direction to be

(13.3.5)

From this we find that the remote coordinate speed of the ship is given by

(13.3.6)

The only speed limitation is

In the ship frame the stress energy tensor of the inner matter shell is given by the solution in section 12.4. Using this to achieve any , we see the ship does achieve faster than c speeds in the remote observer's coordinates.

Recall the Robertson walker solution eqn 8.1.16

and how the definition of a rigid ruler distance eqn 8.1.19

Led to a rigid ruler velocity for galaxies of 8.1.17

v = HL

where for galaxies far enough away this speed exceeds the speed of light. This way of exceeding the speed of light is therefor known to exist in nature. It is more a question of whether we can artificially apply this principle to construct a metric like 13.3.1. Instead of writing the metric as above according to a remote observer's frame let us now apply this perspective to an observer on the ship and write down the metric we would want according to the ship frame as

(13.3.7)

and solve for the Einstein tensor. The real question is whether matter can be manipulated in real nature to correspond closely enough to the stress-energy tensor corresponding to that Einstein tensor in order to yield the desired effects. Essentially this would use the ability of space to expand and contract as the solution for a big bang does in order to bring the destination arbitrarily close to the ship.

Another difficulty in this method is the matter of distinguishing in the corresponding stress-energy tensor what terms are associated with the matter of the surrounding universe Vs which are the ship responsible for corresponding to the warp drive effects. For example without the ship arranging matter into a warp drive and if the ships overall mass can be neglected the metric you want this to reduce to must be a frame transformation of the Robertson-Walker metric which is not vacuum.

Exercises

Problem 13.3.1

Verify that Eqn. 13.3.3 is a vacuum field solution by showing that it is a transformation of the metric of the theory of special relativity.

Problem 13.3.2

Lets say a ship can reach . If T(z = 0) = 25, what does Z(z = 0) have to be for a remote coordinate ship speed of 10c?